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\begin{document}
\centerline{Math 201 Homework for Tuesday, Week 13}
\bigskip
\textbf{Due:} Tuesday, November 30.

\begin{prob}
  Consider the inner product space $(\R^2, \langle \, , \, \rangle)$ presented as an example in class, where the inner product is defined as
  \[\langle(x_1,x_2),(y_1,y_2)\rangle=3x_1y_1+2x_1y_2+2x_2y_1+4x_2y_2.\]
  \begin{enumerate}
   \item Compute the lengths of the vectors $(1,1)$ and $(1,-1)$.
   \item Compute the cosine of the angle between $(1,1)$ and $(1,-1)$. Are these vectors perpendicular?
   \item Find a non-zero vector perpendicular to $(1,1)$.
  \end{enumerate}
\end{prob}

\begin{prob}
  Recall the inner product defined on $M_{m\times n}(F)$, where $F=\R$ or $\C$: for $A,B\in M_{m\times n} (F)$, we define
  \[
    \langle A , B \rangle = \tr(B^\ast A),
  \]
  where $B^\ast = \overline{B^t}$ is the conjugate transpose. In this problem we will verify that this function does indeed satisfy the axioms of an inner product.
  \begin{enumerate}
    \item Prove that this function is  linear in the first coordinate: for all $A,B,C\in M_{m\times n}(F)$ and $r\in F$,
      \[
	\langle A+rC,B\rangle=\langle A,B\rangle + r\langle C,B\rangle.
      \]
    \item Prove that this function is conjugate symmetric: for all $A,B \in M_{m\times n}(F)$,
      \[
	\langle A,B \rangle = \overline{\langle B, A \rangle}.
      \]
    \item Prove that this function is  positive-definite: for all $A \in M_{m\times n} (F)$ with $A\neq 0$,
      \[
	\langle A , A \rangle > 0.
      \]
  \end{enumerate}
\end{prob}

\begin{prob}
  Let $(V, \langle \, , \, \rangle)$ be an inner product space over $\R$, and let $v,w \in V$ be nonzero vectors. 
    \begin{enumerate}
  \item Prove that if the vector
  \[\frac{v}{\|v\|}+\frac{w}{\|w\|}\]
  is nonzero, then it bisects the angle between $v$ and $w$. 
  \item Illustrate in $\R^2$ with the standard dot product.
  \end{enumerate}
\end{prob}
\end{document}


